FourierMatrix - Revision history

Nathaniel at 13:33, 24 June 2015

2015-06-24T13:33:17Z

← Older revision		Revision as of 13:33, 24 June 2015
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	<tt>'''FourierMatrix'''</tt> is a [[List of functions\|function]] that returns the unitary matrix that implements the [http://en.wikipedia.org/wiki/Quantum_Fourier_transform quantum Fourier transform]. That is, it returns the $d \times d$ matrix		<tt>'''FourierMatrix'''</tt> is a [[List of functions\|function]] that returns the unitary matrix that implements the [http://en.wikipedia.org/wiki/Quantum_Fourier_transform quantum Fourier transform]. That is, it returns the $d \times d$ matrix
	: <math>\frac{1}{\sqrt{d}}\begin{bmatrix}1 & 1 & 1 & \cdots & 1 \\ 1 & \omega & \omega^2 & \cdots & \omega^{d-1} \\ 1 & \omega^2 & \omega^4 & \cdots & \omega^{2(d-1)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)(d-1)} \end{bmatrix},</math>		: <math>\frac{1}{\sqrt{d}}\begin{bmatrix}1 & 1 & 1 & \cdots & 1 \\ 1 & \omega & \omega^2 & \cdots & \omega^{d-1} \\ 1 & \omega^2 & \omega^4 & \cdots & \omega^{2(d-1)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)(d-1)} \end{bmatrix},</math>
	where $\omega := \exp(2\pi i/d)$ is a [http://en.wikipedia.org/wiki/Root_of_unity primitive $d$-th root of unity].		where $\omega := \exp(2\pi i/d)$ is a [http://en.wikipedia.org/wiki/Root_of_unity primitive d-th root of unity].

	==Syntax==		==Syntax==
	* <tt>F = FourierMatrix(DIM)</tt>		* <tt>F = FourierMatrix(DIM)</tt>

Nathaniel: Removed red links and expanded intro

2014-12-20T04:14:20Z

Removed red links and expanded intro

← Older revision		Revision as of 04:14, 20 December 2014
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	\|name=FourierMatrix		\|name=FourierMatrix
	\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]		\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]
	\|desc=Generates the [[unitary matrix]] that implements the [[quantum Fourier transform]]		\|desc=Generates the unitary matrix that implements the quantum Fourier transform
	\|upd=November 30, 2012		\|upd=November 30, 2012}}
	~~\|v=0.50~~}}		<tt>'''FourierMatrix'''</tt> is a [[List of functions\|function]] that returns the unitary matrix that implements the [http://en.wikipedia.org/wiki/Quantum_Fourier_transform quantum Fourier transform]. That is, it returns the $d \times d$ matrix
	<tt>'''FourierMatrix'''</tt> is a [[List of functions\|function]] that returns the [[unitary matrix]] that implements the [[quantum Fourier transform]].		: <math>\frac{1}{\sqrt{d}}\begin{bmatrix}1 & 1 & 1 & \cdots & 1 \\ 1 & \omega & \omega^2 & \cdots & \omega^{d-1} \\ 1 & \omega^2 & \omega^4 & \cdots & \omega^{2(d-1)} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)(d-1)} \end{bmatrix},</math>
			where $\omega := \exp(2\pi i/d)$ is a [http://en.wikipedia.org/wiki/Root_of_unity primitive $d$-th root of unity].
	==Syntax==		==Syntax==
	* <tt>F = FourierMatrix(DIM)</tt>		* <tt>F = FourierMatrix(DIM)</tt>
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	==Examples==		==Examples==
	===The qubit Fourier matrix===		===The qubit Fourier matrix===
	The qubit Fourier matrix is simply the usual [[Hadamard gate]]:		The qubit Fourier matrix is simply the usual Hadamard gate:
	<syntaxhighlight>		<syntaxhighlight>
	>> FourierMatrix(2)		>> FourierMatrix(2)

Nathaniel at 15:12, 29 September 2014

2014-09-29T15:12:15Z

← Older revision		Revision as of 15:12, 29 September 2014
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	{{Function		{{Function
	\|name=FourierMatrix		\|name=FourierMatrix
			\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]
	\|desc=Generates the [[unitary matrix]] that implements the [[quantum Fourier transform]]		\|desc=Generates the [[unitary matrix]] that implements the [[quantum Fourier transform]]
	\|upd=November 30, 2012		\|upd=November 30, 2012
	\|v=1.00}}		\|v=0.50}}
	<tt>'''FourierMatrix'''</tt> is a [[List of functions\|function]] that returns the [[unitary matrix]] that implements the [[quantum Fourier transform]].		<tt>'''FourierMatrix'''</tt> is a [[List of functions\|function]] that returns the [[unitary matrix]] that implements the [[quantum Fourier transform]].

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	===The qubit Fourier matrix===		===The qubit Fourier matrix===
	The qubit Fourier matrix is simply the usual [[Hadamard gate]]:		The qubit Fourier matrix is simply the usual [[Hadamard gate]]:
	<~~pre~~>		<syntaxhighlight>
	>> FourierMatrix(2)		>> FourierMatrix(2)

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	0.7071 0.7071		0.7071 0.7071
	0.7071 -0.7071 + 0.0000i		0.7071 -0.7071 + 0.0000i
	</~~pre~~>		</syntaxhighlight>

	===The three-qubit Fourier matrix===		===The three-qubit Fourier matrix===
	The following line of code generates the three-qubit (i.e., <tt>DIM = 8</tt>) Fourier matrix, which can be seen [http://en.wikipedia.org/wiki/Quantum_Fourier_transform here]. The multiplication by <tt>sqrt(8)</tt> is just there to make the output easier to read.		The following line of code generates the three-qubit (i.e., <tt>DIM = 8</tt>) Fourier matrix, which can be seen [http://en.wikipedia.org/wiki/Quantum_Fourier_transform here]. The multiplication by <tt>sqrt(8)</tt> is just there to make the output easier to read.
	<~~pre~~>		<syntaxhighlight>
	>> FourierMatrix(8)*sqrt(8)		>> FourierMatrix(8)*sqrt(8)

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	1.0000 -0.0000 - 1.0000i -1.0000 + 0.0000i 0.0000 + 1.0000i 1.0000 - 0.0000i -0.0000 - 1.0000i -1.0000 + 0.0000i 0.0000 + 1.0000i		1.0000 -0.0000 - 1.0000i -1.0000 + 0.0000i 0.0000 + 1.0000i 1.0000 - 0.0000i -0.0000 - 1.0000i -1.0000 + 0.0000i 0.0000 + 1.0000i
	1.0000 0.7071 - 0.7071i -0.0000 - 1.0000i -0.7071 - 0.7071i -1.0000 + 0.0000i -0.7071 + 0.7071i 0.0000 + 1.0000i 0.7071 + 0.7071i		1.0000 0.7071 - 0.7071i -0.0000 - 1.0000i -0.7071 - 0.7071i -1.0000 + 0.0000i -0.7071 + 0.7071i 0.0000 + 1.0000i 0.7071 + 0.7071i
	</~~pre~~>		</syntaxhighlight>

			{{SourceCode\|name=FourierMatrix}}

Nathaniel: Created page with "{{Function |name=FourierMatrix |desc=Generates the unitary matrix that implements the quantum Fourier transform |upd=November 30, 2012 |v=1.00}} `'''FourierMatrix''..."`

2012-12-01T05:38:42Z Created page with "{{Function |name=FourierMatrix |desc=Generates the unitary matrix that implements the quantum Fourier transform |upd=November 30, 2012 |v=1.00}} <tt>'''FourierMatrix''..." New page {{Function |name=FourierMatrix |desc=Generates the [[unitary matrix]] that implements the [[quantum Fourier transform]] |upd=November 30, 2012 |v=1.00}} <tt>'''FourierMatrix'''</tt> is a [[List of functions|function]] that returns the [[unitary matrix]] that implements the [[quantum Fourier transform]]. ==Syntax== * <tt>F = FourierMatrix(DIM)</tt> ==Argument descriptions== * <tt>DIM</tt>: The dimension of the system. In other words, <tt>F</tt> will be a <tt>DIM</tt>-by-<tt>DIM</tt> matrix. ==Examples== ===The qubit Fourier matrix=== The qubit Fourier matrix is simply the usual [[Hadamard gate]]: <pre> >> FourierMatrix(2) ans = 0.7071 0.7071 0.7071 -0.7071 + 0.0000i </pre> ===The three-qubit Fourier matrix=== The following line of code generates the three-qubit (i.e., <tt>DIM = 8</tt>) Fourier matrix, which can be seen [http://en.wikipedia.org/wiki/Quantum_Fourier_transform here]. The multiplication by <tt>sqrt(8)</tt> is just there to make the output easier to read. <pre> >> FourierMatrix(8)*sqrt(8) ans = 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.7071 + 0.7071i 0.0000 + 1.0000i -0.7071 + 0.7071i -1.0000 + 0.0000i -0.7071 - 0.7071i -0.0000 - 1.0000i 0.7071 - 0.7071i 1.0000 0.0000 + 1.0000i -1.0000 + 0.0000i -0.0000 - 1.0000i 1.0000 - 0.0000i 0.0000 + 1.0000i -1.0000 + 0.0000i -0.0000 - 1.0000i 1.0000 -0.7071 + 0.7071i -0.0000 - 1.0000i 0.7071 + 0.7071i -1.0000 + 0.0000i 0.7071 - 0.7071i 0.0000 + 1.0000i -0.7071 - 0.7071i 1.0000 -1.0000 + 0.0000i 1.0000 - 0.0000i -1.0000 + 0.0000i 1.0000 - 0.0000i -1.0000 + 0.0000i 1.0000 - 0.0000i -1.0000 + 0.0000i 1.0000 -0.7071 - 0.7071i 0.0000 + 1.0000i 0.7071 - 0.7071i -1.0000 + 0.0000i 0.7071 + 0.7071i -0.0000 - 1.0000i -0.7071 + 0.7071i 1.0000 -0.0000 - 1.0000i -1.0000 + 0.0000i 0.0000 + 1.0000i 1.0000 - 0.0000i -0.0000 - 1.0000i -1.0000 + 0.0000i 0.0000 + 1.0000i 1.0000 0.7071 - 0.7071i -0.0000 - 1.0000i -0.7071 - 0.7071i -1.0000 + 0.0000i -0.7071 + 0.7071i 0.0000 + 1.0000i 0.7071 + 0.7071i </pre>

FourierMatrix - Revision history

Nathaniel at 13:33, 24 June 2015

Nathaniel: Removed red links and expanded intro

Nathaniel at 15:12, 29 September 2014

Nathaniel: Created page with "{{Function |name=FourierMatrix |desc=Generates the unitary matrix that implements the quantum Fourier transform |upd=November 30, 2012 |v=1.00}} '''FourierMatrix''..."

Nathaniel: Created page with "{{Function |name=FourierMatrix |desc=Generates the unitary matrix that implements the quantum Fourier transform |upd=November 30, 2012 |v=1.00}} `'''FourierMatrix''..."`